ON THE GUTTING-OFF OF A BATIO 177 
1523.2 
N 
nts from 
a given 
g on the 
le given 
ght line t 
actively 
Books of 
this pro 
ís of the 
ry condi- 
on is not 
>sing the 
ent cases 
vhich the 
1 or B, is 
ds to the 
oves that 
: I where 
on of the 
ieting AC 
T. Then 
B, is con- 
the ratio 
nethod of 
Ived and 
AM is a 
Draw OG parallel to BN or B'N' to meet AM in 0. Take 
D on AM such that OG :AD = X= B'N' : AM. 
Then AM : AD = B'N' : OG 
= B'M: CM; 
therefore MD : AD = B'G : CM, 
or CM. MD = AD. B'G, a given rectangle. 
Hence the problem is reduced to one of applying to CD a 
rectangle (CM. MD) equal to a given rectangle (AD. B'G) hut 
falling short by a square figure. In the case as drawn, what 
ever be the value of A, the solution is always possible because 
the given rectangle AD . CB' is always less than G A . AD, and 
therefore always less than f GD' 1 ; one of the positions of 
M falls between A and D because GM. MD< G A . AD. 
The proposition III. 41 of the Conics about the intercepts 
made on two tangents to a parabola by a third tangent 
(pp. 155-6 above) suggests an obvious application of our pro 
blem. We had, with the notation of that proposition, 
Pr : rq = rQ : Qp = qp :q>R. 
Suppose that the two tangents qP, qR are given as fixed 
tangents with their points of contact P, R. Then we can 
draw another tangent if we can draw a straight line 
intersecting qP,qR in such a way that Pr : rq — qp : pR or 
Pq-.qr = qR : pR, i. e. qr : pR = Pq : qR (a constant ratio) ; 
i.e. we have to draw a straight line such that the intercept bj?- 
it on qP measured from q has a given ratio to the intercept 
by it on qR measured from R. This is a particular case of 
our problem to which, as a matter of fact, Apollonius devotes 
special attention. In the annexed figure the letters have the 
same meaning as before, and N'M has to be drawn through 0 
such that B'N': AM = A. In this case there are limits to